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Theorem sucexeloni 7780
Description: If the successor of an ordinal number exists, it is an ordinal number. This variation of onsuc 7782 does not require ax-un 7708. (Contributed by BTernaryTau, 30-Nov-2024.) (Proof shortened by BJ, 11-Jan-2025.)
Assertion
Ref Expression
sucexeloni ((𝐴 ∈ On ∧ suc 𝐴𝑉) → suc 𝐴 ∈ On)

Proof of Theorem sucexeloni
StepHypRef Expression
1 eloni 6363 . . 3 (𝐴 ∈ On → Ord 𝐴)
2 ordsuci 7779 . . 3 (Ord 𝐴 → Ord suc 𝐴)
31, 2syl 17 . 2 (𝐴 ∈ On → Ord suc 𝐴)
4 elex 3491 . 2 (suc 𝐴𝑉 → suc 𝐴 ∈ V)
5 elong 6361 . . 3 (suc 𝐴 ∈ V → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
65biimparc 480 . 2 ((Ord suc 𝐴 ∧ suc 𝐴 ∈ V) → suc 𝐴 ∈ On)
73, 4, 6syl2an 596 1 ((𝐴 ∈ On ∧ suc 𝐴𝑉) → suc 𝐴 ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  Vcvv 3473  Ord word 6352  Oncon0 6353  suc csuc 6355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6356  df-on 6357  df-suc 6359
This theorem is referenced by:  onsuc  7782  1on  8460  2on  8462
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